Since \(\Gamma , \Gamma _n \in {B}_0({H})\subseteq {B}({H})\)  then it follows that the resolvent \(R(z ; \Gamma ) := (z - \Gamma )^{-1}\)  is also in \({B}({H})\) , and thus \((\Gamma - \Gamma _n)(z - \Gamma )^{-1} \in {B}({H})\) . Hence we can bound the operator norm \(\vert \vert \cdot \vert \vert \)  by the Hilbert-Schmidt norm \(\vert \vert \cdot \vert \vert _{\mathrm {HS}}\) 
See [1]}
\(\vert \vert (\Gamma - \Gamma _n)(z - \Gamma )^{-1}\vert \vert ^2&\le \vert \vert (\Gamma - \Gamma _n)(z - \Gamma )^{-1}\vert \vert ^2_{\mathrm {HS}} \nonumber \\&\equiv \sum _{l = 1}^\infty \vert \vert  (\Gamma - \Gamma _n)(z - \Gamma )^{-1}(e_l) \vert \vert ^2 \nonumber \\&= \sum _{l, k = 1}^\infty \frac{1}{(z - \lambda _l)^2} \left| \left\langle (\Gamma - \Gamma _n)e_l,e_k \right\rangle  \right|^2\) 
